Theorem zfext2 1087

Description: A generalization of the Axiom of Extensionality in which x and y need not be distinct.

Assertion

Ref	Expression
zfext2	|- (A.z(z e. x <-> z e. y) -> x = y)

Distinct variable group(s):   x,z   y,z

Proof of Theorem zfext2

Step	Hyp	Ref	Expression
1		a9e 809	. 2 |- E.w w = x
2		ax-ext 1074	. . . 4 |- (A.z(z e. w <-> z e. y) -> w = y)
3		a14b 820	. . . . . . 7 |- (w = x -> (z e. w <-> z e. x))
4	3	bibi1d 471	. . . . . 6 |- (w = x -> ((z e. w <-> z e. y) <-> (z e. x <-> z e. y)))
5	4	bialdv 935	. . . . 5 |- (w = x -> (A.z(z e. w <-> z e. y) <-> A.z(z e. x <-> z e. y)))
6		a8b 817	. . . . 5 |- (w = x -> (w = y <-> x = y))
7	5, 6	imbi12d 474	. . . 4 |- (w = x -> ((A.z(z e. w <-> z e. y) -> w = y) <-> (A.z(z e. x <-> z e. y) -> x = y)))
8	2, 7	mpbii 168	. . 3 |- (w = x -> (A.z(z e. x <-> z e. y) -> x = y))
9	8	19.23aiv 952	. 2 |- (E.w w = x -> (A.z(z e. x <-> z e. y) -> x = y))
10	1, 9	ax-mp 6	1 |- (A.z(z e. x <-> z e. y) -> x = y)

Syntax hints:   -> wi 2   <-> wb 127  A.wal 672  E.wex 678   = weq 797   e. wel 803

This theorem is referenced by:  axextnd 3737

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-8 798  ax-9 799  ax-12 802  ax-14 805  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679

metamath.org